Woodworking Math Tables, Formulas and Calculators - Part III
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Polygon Formulas, Calculator and Tables

On the page you'll find formulae for calculating the dimensions and angles necessary to build multi-sided shapes and tables containing dimensions and angles for common sizes of polygons.

Polygon Formulas

Formulas for calculating dimensions and angles used in building regular polygon shapes are listed below.  Please be sure to double-check your calculations carefully and make an appropriate number of test cuts before you begin sawing the stock you've invested in for your project.

N = Number of sides of the polygon
B = 180/N
(See Note 1 below)
S = sin(B) x D1
S = tan(B) x D2
D1 = S/sin(B)
D1 = D2 /cos(B)
D2 = S/tan(B)
D2 = cos(B) x D1
D4 = D2 - (material thickness x 2)
D3 = D4 / cos(B)M = D2 - D3
(see Note 2 below)

Note 1: "B" is the bevel or miter angle setting for saws that calibrate a square cut at zero degree.  For saws that calibrate a square cut at 90 degrees, use the complement of B for the saw's setting.

Note 2: Dimension "M" is useful in projects that involve rounding off the corners of a polygon to form a round or cylindrical shape. Dimension M will tell you the thickness of the material where the staves or sides of the shape are joined.

Tables

Use the quick reference tables below to double-check angles and dimensions of common sizes of polygons.

Polygon Angles and Dimensions
Number of Sides Bevel
Angle
Side-to-Side Width Tip-to-Tip Width Side Length
Decimal Approx. Fractional Decimal Approx. Fractional Decimal Approx. Fractional
6 30° 5.1962 5-1164 6.0 6 3.0 3
6.0 6 6.9282 6-59/64 3.4641 3-15/32
6.9282 6-63/64 8.0 8 4.0 4
8.0 8 9.2376 9-15/64 4.6188 4-5/8
8.6603 8-21/32 10.0 10 5.0 5
10.0 10 11.5470 11-35/64 5.7735 5-25/32
10.3923 10-25/64 12.0 12 6.0 6
12.0 12 13.8564 13-55/64 6.9282 6-59/64
12.1244 12-1/8 14.0 14 7.0 7
14.0 14 16.1658 16-11/64 8.0829 8-5/64
13.8564 13-55/64 16.0 16 8.0 8
16.0 16 18.4752 18-15/32 9.2376 9-15/64
15.5885 15-19/32 18.0 18 9.0 9
18.0 18 20.7846 20-25/32 10.3923 10-25/32
Side-to-Side Width Tip-to-Tip Width Side Length
Number of Sides Bevel
Angle
Decimal Fractional
Equivalent
Decimal Fractional
Equivalent
Decimal Fractional
Equivalent
8 22.5° 5.5433 5-35/64 6.0 6 2.2961 2-19/64
6.0 6 6.4944 6-1/2 2.4853 2-31/64
7.391 7-25/64 8.0 8 3.0615 3-1/16
8.0 8 8.6591 8-21/32 3.3137 3-5/16
9.2388 9-15/64 10.0 10 3.8268 3-53/64
10.0 10 10.8239 10-53/64 4.1421 4-9/64
11.0866 12.0 12 4.5922 4-19/32
12.0 12 12.9887 13 4.9706 4-31/32
12.9343 12-15/16 14.0 14 5.3576 5-23/64
14.0 14 15.1535 15-5/32 5.799 5-51/64
14.7821 14-25/32 16.0 16 6.1229 6-5/8
16.0 16 17.3183 17-5/16 6.6274 6-5/8
16.6298 16-5/8 18.0 18 6.8883 6-57/64
18.0 18 19.483 19-31/64 7.4558 7-29/64
Side-to-Side Width Tip-to-Tip Width Side Length
Number of Sides Bevel
Angle
Decimal Fractional
Equivalent
Decimal Fractional
Equivalent
Decimal Fractional
Equivalent
12 15° 5.7956 5-51/64 6.0 6 1.5529 1-35/64
6.0 6 6.2117 6-7/32 1.6077 1-39/64
7.7274 7-47/64 8.0 8 2.0706 2-5/64
8.0 8 8.2822 8-9/32 2.1436 2-9/64
9.6593 9-21/32 10.0 10 2.5882 2-19/32
10.0 10 10.3528 10-23/64 2.6795 2-43/64
11.5911 11-19/32 12.0 12 3.1058 3-7/64
12.0 12 12.4233 12-27/64 3.2154 3-7/32
13.523 13-33/64 14.0 14 3.6235 3-5/8
14.0 14 14.4939 14-1/2 3.7513 4-3/4
15.4548 15-29/64 16.0 16 4.1411 4-9/64
16.0 16 16.5644 16-9/16 4.2872 4-9/32
17.3867 17-25/64 18.0 18 4.6587 4-21/32
18.0 18 18.635 18-41/64 4.8231 4-53/64
Side-to-Side Width Tip-to-Tip Width Side Length
Number of Sides Bevel
Angle
Decimal Fractional
Equivalent
Decimal Fractional
Equivalent
Decimal Fractional
Equivalent
16 11.25 5.8847 5-57/64 6.0 6 1.1705 1-11/64
6.0 6 6.1176 6-5/8 1.1935 1-3/16
7.8463 7-27/32 8.0 8 1.5607 1-9/16
8.0 8 8.1567 8-5/32 1.5913 1-19/32
9.8079 9-13/16 10.0 10 1.9509 1-61/64
10.0 10 10.1959 10-13/64 1.9891 2
11.7694 11-49/64 12.0 12 2.3411 2-11/32
12.0 12 12.2351 12-15/64 2.3869 2-25/64
13.731 13-47/64 14.0 14 2.7313 2-47/64
14.0 14 14.2743 14-9/32 2.7848 2-25/32
15.6926 15-11/16 16.0 16 3.1214 3-1/8
16.0 16 16.3135 16-5/16 3.1826 3-3/16
17.6541 17-21/32 18.0 18 3.5116 3-33/64
18.0 18 18.3526 18-23/64 3.5804 3-37/64
posted on August 15, 2013 by Rockler
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Comments

9 thoughts on “Woodworking Math Tables, Formulas and Calculators - Part III”

  • John Sterba

    Rockler is my favorite go to for accurate info for advice on my projects. i use them when working with all substrates.

  • amir hossein azimi
    amir hossein azimi June 17, 2014 at 10:22 am

    hello

  • Bob

    What happened to the calculator that used to be here?

  • E. C. Bearden

    What happened to the calculator that was on this page? Please send me a link to it if it resides at another location. I am willing to pay a small fee for its use. Thank you.
    E. C. Bearden

  • ec bearden

    what happened to the calculator on this page?

  • Stan

    Working on our favourite workshop materials I appreciate all the angles, polygons etc. one often needs. However, for drawing precise circles on these surfaces I've found our kitchen crockery, even great ovals, in different sized plates and dishes supplying all I need.

  • HenryF

    In the interest of sparing fellow woodworkers from confusion, I'd like to point out that the last equation listed above—
    “ D3 = D4 / cos(B)M = D2 - D3 "
    —has a glitch in it; since to have D3 = D2 - D3,
    D2 would have to always equal 2(D3).

    As the drawing shows, D2 = D3 + 2M,
    so, M would always have to be 1/2 of D3
    which is not what the equations set out to describe.
    I hope this is helpful, or at least constructively confusing.

  • Glen W. Petrie

    This is what I came up for the calculations you show here.

    Given:
    N : number of segments
    D2 : outer diameter
    M : wall thickness

    Bo = 180 / N
    S = D2 tan(Bo)
    D1 = D2 / cos(Bo)
    D3 = D2 – 2 M
    D4 = D3 cos(Bo)
    m = (D1 – D3) / 2
    s = D4 tan(Bo)


    m = distance of side of cut segments
    s = length on the inside of segments

    I can send you a diagram if you like

  • Perry Thompson
    Perry Thompson March 1, 2016 at 2:49 pm

    You guys make this way to.complicated ! Simply put to find your angle of you cut ,
    Simply divide the quantity of peices into 180 . Example , 18 pcs equals 10 degree cut . Now you need the length of the cut , find the diameter and divide into the amount of peices or parts, into the Circumference and yo have t length . This work on all circles etc. Professional woodturner 40 years

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